In a medium saucepan over medium-high heat, bring water and salt to a boil. Add the cornmeal in a slow, steady stream, whisking during this process. Whisk in 1 tablespoon olive oil and thyme. Reduce heat to maintain a simmer, and cook, stirring frequently until mixture is very thick, about 15 to 20 minutes. Stir in 3 tablespoons Parmesan cheese.

Meanwhile, heat remaining 1 tablespoon olive oil in a large nonstick skillet, over medium-high heat. Add the mushrooms and cook, stirring frequently until the liquid boils away and the mushrooms are golden. …